First-Principles Study of the Structural, Mechanical and Thermodynamic Properties of Al₁₁RE₃ in Aluminum Alloys

Abstract

The stability and mechanical and thermodynamic properties of Al₁₁RE₃ intermetallics (RE = Sc, Y and lanthanide La-Lu) have been investigated by combining first-principles and Debye model calculations. It was found that the formation enthalpies of the Al₁₁RE₃ intermetallics are all negative, indicating that they are stable; moreover, the experimental values of Al₁₁La₃ and Al₁₁Ce₃ are in good agreement with the predicted values, which are −0.40 kJ/mol and −0.38 kJ/mol, respectively. The calculated results of the mechanical properties reveal that the Young’s modulus E and shear modulus G of Al₁₁RE₃ (RE = La, Ce, Pr, Nd and Sm) intermetallics are obviously greater than that of Al, implying that the stiffness, toughness, and tensile strength of them are significantly greater than those of aluminum, and that they, as strengthen phases, can effectively improve the mechanical property of aluminum alloys. The Poisson’s ratio v of Al₁₁Sc₃ (0.37) is the largest, and the heterogeneity is obvious. All the Al₁₁RE₃ intermetallics can enhance the thermostability of the aluminum because of their lower Gibbs free energy F in the range of −5.002~−4.137 eV/atom and thermal expansion coefficient α of Al in the range of 2.34~2.89 × 10⁻⁵/K at 300K, as well as higher entropy and constant volume-specific heat than aluminum at finite temperatures. With an increase in the atomic number, different change trends were observed for the formation enthalpy ΔH_f, bulk modulus B, Young’s modulus E, and shear modulus G. This paper can provide ideas and help for designing a high-performance, heat-resistant aluminum alloy.

1. Introduction

Aluminum (Al) alloys have become an important lightweight material for energy conservation and emission reduction in the automobile, rail transit, aerospace, and other industries due to their many excellent performances, including their light weight, high specific strength, good formability, excellent weldability, and corrosion resistance [1,2,3,4,5,6]. As engineering materials, Al alloys have been widely used at room temperature; however, it is difficult to meet the requirements of high-temperature devices, such as aircraft and automobile engines, that have been used for a long time at more than 200 °C [7,8]. The cause of this issue is the rapid coarsening and dissolution of the strengthen phases in the Al alloys at high temperature, which leads to a sharp decrease in the mechanical properties of the alloys [9,10]. Therefore, improving the thermal stability of the strengthen phases at high temperatures is the key to expanding the application of Al alloys in high-temperature devices.

In recent years, alloying by adding rare-earth elements (RE) has become an important and effective method to improve the high-temperature performance of the Al alloys, mainly because of the formation of the high temperature-resistant strengthen phases and Al_xRE_y intermetallics in the alloy, such as the L1₂-Al₃RE (e.g., RE = Sc, Er) [11,12,13,14] and Al₁₁RE₃ (e.g., RE = La, Ce) intermetallics [15,16,17,18]. It is widely known that the physical and mechanical properties of alloys mainly depend on the basic properties of the strengthen phases. Consequently, the research hotspots about Al₁₁RE₃ alloys focus on the basic properties of the Al_xRE_y intermetallics, and the relevant research results have been obtained through experimentation.

Nevertheless, through experimentation, an investigation of all the Al_xRE_y intermetallics would require the synthesis, processing, and characterization of many Al₁₁RE₃ alloys, making it complex, inefficient, and very expensive. Instead, the theoretical identification, e.g., first-principles calculations, which plays a very important role in materials research, particularly in complicated systems, can make this work easier and cheaper because of the development of modern computer technology [19,20,21,22].

In fact, there have been systematic studies on the basic properties of first-principles calculations, including on the stability, elastic parameters, and thermal properties, such as free energy, entropy, constant volume heat capacity, and the volume thermal expansion of the L1₂-Al₃RE strengthen phases in Al alloys [23,24,25]. However, to our best knowledge, no systematic first-principles computational studies have been performed on Al₁₁RE₃ intermetallics. In this work, we have used the first-principles calculations combined with quasi-harmonic approximation to investigate the structural, elastic, thermal, and electrical properties of the Al₁₁RE₃ (Sc, Y, and lanthanide La-Lu) intermetallics in Al-based alloys at various temperatures and pressures. The lattice constant, lattice mismatch, and formation enthalpy of Al₁₁RE₃ compounds are firstly calculated, and the underlying mechanisms are probed. Then, the thermodynamic parameters of Al₁₁RE₃ compound are calculated using a Debye model to understand the high temperature performance of aluminum alloys.

2. Computation Detail

First-principles calculations based on density functional theory (DFT) [26] employed the plane-wave, total-energy methodology, which solved the Kohn–Sham equation to obtain the valence electron density and the wave function as implemented in the Vienna ab initio Simulation Package (VASP) [27]. The interaction between the ionic core and valence electrons [28] was described by the Projector Augmented Wave (PAW) [29] method, and the exchange correlation function was calculated using the Perdew–Burke–Ernzerhof (PBE) version of the generalized gradient approximation (GGA) [30]. To ensure that the results are accurate, the cutoff energy of the plane-wave basis was chosen as 350 eV, and Brillouin zone sampling was performed using the Gamma-centered Monkhorst–Pack method with 8 × 3 × 3 k–mesh grids [31]. To obtain a better calculation and accurate results, the ground state geometry was relaxed using conjugate gradient (CG) minimization, and the cell shape and internal atomic coordinates were completely relaxed until the electron self-consistent convergence criterion of the electron iteration was 10⁻⁶ eV/atom and the Hellmann–Feynman force [32] was less than 0.01 eV/Å. In the calculation process, the PAW_GGA pseudopotentials of Al, Sc, La, Y_sv, and (Ce-Lu)_3 are used to simulate the size of the real potential. The energy–strain method was used for the mechanical calculation, the VASPKIT code [33] was used to fit the equation of state (EOS), and the strain values were −9% ~9%. The thermodynamic properties were calculated by establishing the Debye–Grüneisen model [34,35] and using the Gibbs2 code [36]. Although it is possible to calculate the thermodynamic properties of the material using the quasi harmonic approximation model (QHA), it is still a complex and expensive task to calculate the full vibration spectrum of the crystal on the volume grid. At the same time, compared with a fixed Poisson’s ratio v in Debye–Grüneisen model, the exact Poisson’s ratio v of each alloy is calculated using energy strain method in this experiment, which further improves the accuracy on the basis of the original. However, the Debye model only considers the contribution of the long wave part but not the short wave part, which makes the error of calculation at a high temperature larger. At the same time, the Al₁₁RE₃ unit cell model required for the calculation was derived by the program software of VESTA to visualize the 3D structure, as shown in Figure 1a. It can be seen from the following figure that the structure belongs to the orthorhombic crystal system.

3. Result and Discussion

3.1. Enthalpy of Formation and Stability

For the purpose of evaluating the relative stabilities of the Al₁₁RE₃ intermetallics in Al alloys, we calculated the formation enthalpy ΔH_f according to the following equation [37]

$$\Delta H_f=\frac{E_{tot}-x{E}_{Al}-y{E}_{RE}}{x+y}$$

(1)

where $E_{tot}$ is the total energy of different compounds; $E_{Al}$ and $E_{RE}$ are the calculated total energy of the elements of Al and RE in a steady state, respectively. Parameters x and y are the number of Al and RE atoms in the unit cell, respectively.

Based on the calculated formation enthalpy ΔH_f of Al₁₁RE₃ intermetallics and the previous theoretical data [20] at 0 K and 0 GPa, as shown in Figure 1b, we can obtain the following conclusions: (1) The calculated values were in good agreement with the previous theoretical predictions, which indicates that the present calculations are accurate and reliable; (2) the formation enthalpies of the Al₁₁RE₃ intermetallics are all negative, indicating that they are stable compounds; (3) with an increase in the atomic number, the formation enthalpy decreases first and then increases. The maximum and minimum values of the ΔH_f appear at the Sc and La elements, respectively, which indicates that the thermodynamic stability of Al₁₁La₃ is the strongest, followed by Al₁₁Ce₃, while the thermodynamic stability of Al₁₁Sc₃ is the worst. These results are in good agreement with the experimental findings that Al₁₁La₃ and Al₁₁Ce₃ particles are the most found compounds in the experiment [38].

In this experiment, the formation enthalpy $\mathsf{\Delta }{H}_f$ of the aluminum alloy Al₁₁RE₃ doped with 11 rare earth elements was calculated according to the formula. Compared with the existing experimental values [20], as shown in Figure 2, it can be clearly seen that the existing experimental values of Al₁₁La₃ and Al₁₁Ce₃ are in good agreement with the existing theoretical predicted values, which are −0.40 kJ/mol and −0.38 kJ/mol, respectively. We can observe that the formation enthalpies of the binary aluminum alloys are all negative, indicating that there are exothermic reactions when they form compounds, and that they are all stable compounds [20]. At the same time, Figure 1b also helps us better understand the relationship between the formation enthalpy $\mathsf{\Delta }{H}_f$ and the atomic number of rare earth elements. We can clearly observe that the overall curve shows a trend that first decreases and then increases. Moreover, the formation enthalpy $\mathsf{\Delta }{H}_f$ reaches the maximum value at the Sc element and then shows a downward trend; the lanthanide rare earth element La reaches a minimum value of −0.40 eV/atom, and the atomic number increases until the Lu element shows a generally upward trend; however, the formation enthalpy $\mathsf{\Delta }{H}_f$ of the Sc element is not exceeded. This shows that the thermodynamic stability of Al₁₁La₃ is the strongest [38], while the thermodynamic stability of Al₁₁Sc₃ is the weakest.

3.2. Elastic Properties

Elastic constants and elastic moduli are important components in understanding the mechanical properties of materials [39]. The elastic constant can be used to determine the mechanical stability of the material, reflect the ability of the crystal to resist external stress [20], and obtain important information on the stability of the aluminum alloy unit cell structure and the anisotropic characteristics of the crystal. By calculating the elastic constant of the alloy [40], the elastic modulus such as bulk modulus B, shear modulus G, Young’s modulus E, and Poisson’s ratio ν can be further determined to determine the mechanical properties of the material. The elastic constant is first calculated from a small strain of a uniform unit cell and is then determined by using the corresponding change in the total energy [41]; finally, the elastic modulus is derived using the Voigt–Reuss–Hill approximation [42].

C_ij is the crystal’s response to an applied external stress, while the number of elastic constants lies in the symmetry of the structure. Al₁₁RE₃ is an orthorhombic structure with nine independent elastic constants ($C_{11},C_{12},C_{13},C_{22},C_{23},C_{33},C_{44},C_{55}, and C_{66}$), whose elastic constant matrix is in the form of 6 × 6. On the basis of the Born criterion [43], according to the necessary and sufficient Born criterion under the simplified orthogonal system [44], the following criteria can be obtained:

$$C_{11}>0, C_{11}{C}_{22}>C_{12}^2$$

(2)

$$C_{11}{C}_{22}{C}_{33}+2C_{12}{C}_{13}{C}_{22}-C_{11}{C}_{23}^2-C_{22}{C}_{13}^2-C_{33}{C}_{12}^2>0$$

(3)

$$C_{44}>0 , C_{55}>0 , C_{66}>0$$

(4)

Table 1. The elastic constant C_ij of Al₁₁RE₃ intermetallics (units: GPa).

Compound	C11	C12	C13	C22	C23	C33	C44	C55	C66
Al11Sc3	99.22	71.22	74.54	110.05	65.24	119.06	29.81	36.21	17.95
Al11Y3	120.17	56.87	62.15	107.78	62.81	122.48	37.68	52.18	51.12
Al11La3	129.53	44.00	52.36	115.74	52.97	114.94	53.59	62.02	56.55
Al11Ce3	128.71	46.13	53.92	116.61	54.39	117.82	47.36	57.64	54.54
Al11Pr3	129.35	47.41	55.74	117.31	56.11	120.19	46.33	57.83	54.93
Al11Nd3	129.12	48.55	57.23	117.25	57.67	121.31	45.19	57.52	55.06
Al11Pm3	128.51	49.71	58.47	116.87	59.17	122.52	44.13	57.03	55.03
Al11Sm3	127.56	50.86	59.50	116.08	60.35	123.12	43.20	56.36	54.69
Al11Eu3	127.00	52.31	61.07	115.20	61.62	123.22	42.01	56.17	54.60
Al11Gd3	125.51	53.43	61.95	113.46	62.39	123.07	40.54	55.41	53.81
Al11Tb3	124.19	54.61	62.72	111.77	63.18	122.77	39.24	54.61	52.76
Al11Dy3	122.83	55.88	63.48	109.92	63.90	122.49	37.87	53.88	51.47
Al11Ho3	120.95	57.45	64.74	108.04	64.96	121.11	36.43	53.09	50.23
Al11Er3	118.82	59.06	66.18	105.53	65.82	119.16	34.70	52.59	49.13
Al11Tm3	117.47	60.86	67.65	103.33	66.59	117.93	33.43	52.59	48.32
Al11Yb3	115.03	63.04	69.11	100.37	67.20	116.02	31.97	51.82	46.91
Al11Lu3	111.69	65.12	70.84	96.90	67.55	113.29	30.53	50.89	45.44

and Figure 2 present the relevant elastic constants and elastic moduli, respectively, which are used to determine the mechanical stability of aluminum alloys and the mechanical properties of materials.

Hill proved that the equations obtained using the Voigt–Reuss description of the bulk modulus B and the shear modulus G are the upper and lower limits of the true crystal constant, respectively, which is exactly the arithmetic mean provided by Voigt and Reuss; moreover, the following obtains the orthorhombic crystal formulas for calculating the elastic modulus and shear modulus [42]

$$B=\frac{B_R+B_V}{2}$$

(5)

$$G=\frac{G_R+G_V}{2}$$

(6)

$$B_R=\frac{N^2}{M}$$

(7)

$$B_V=\frac{2\left(C_{11}+C_{12}\right)+C_{33}+4C_{13}}{9}$$

(8)

$$G_R=\frac{15}{18(B_v/N^2)+\left[6/\left(C_{11}-C_{12}\right)\right]+\left(6/C_{44}\right)+\left(3/C_{66}\right)}$$

(9)

$$G_v=\frac{M+3C_{11}-3C_{12}+12C_{44}+C_{66}}{30}$$

(10)

where

$$M=C_{11}+C_{12}+2C_{33}-4C_{13}$$

(11)

$$N^2=\left(C_{11}+C_{12}\right)C_{33}-2C_{13}^2$$

(12)

Using the elastic constants, Young’s modulus E and Poisson’s ratio ν can be expressed as:

$$E=\frac{9GB}{3B+G}$$

(13)

$$v=\frac{3B-2G}{2\left(3B+G\right)}$$

(14)

According to the criterion of mechanical stability and the calculated value of the elastic constant, we can observe from Figure 2a that their values are positive, and that all the rare earth–aluminum alloys meet the conditions of mechanical stability, indicating that they are stable. We can also see from Figure 2a that there are two trends in relation to the La element, as the node shows a trend of first increasing and then decreasing, or first decreasing and then increasing.

Bulk modulus B can be used to measure the average atomic bond strength of a material and is inversely proportional to compressibility; the greater the bulk modulus, the greater the resistance of the material to compression [45]. Observing the bulk modulus B of Al₁₁RE₃ and aluminum in Figure 2b, it is found that the bulk modulus of all aluminum rare earth alloys except Al₁₁Sc3 (83.21 GPa) is smaller than that of Al (81.71 GPa), of which Al₁₁La₃ has the smallest bulk modulus B, and that the minimum is 73.15 GPa. This shows that Al₁₁Sc₃ has the strongest compressive ability, the most stable structure, is not easily deformed, has the strongest rigidity and the highest hardness. On the contrary, Al₁₁La₃ is easy to deform, weak in rigidity, and is the lowest in hardness.

Young’s modulus E and shear modulus G are the most important parameters for evaluating the strength of materials. Young’s modulus E can represent the ability to resist deformation; the larger the value, the harder the material. In addition, the shear modulus G can represent the ability to resist shear stress, and the larger the shear modulus, the more obvious the directional bonds between atoms [45]. It can be seen from Figure 2b that the shear modulus G and Young’s modulus E of Al₁₁RE₃ have the same trend, with La atoms as the nodes, of first rising and then decreasing with an increase in the atomic number. With Al₁₁Sc₃ as the starting point, Young’s modulus E increases from 64.00 GPa to 115.81 GPa for Al₁₁La₃ and then slowly decreases to Al₁₁Lu₃ (81.91 GPa) with an increase in the atomic number. Similarly, the shear modulus G also showed the same trend, rising from 23.33 GPa at Al₁₁Sc₃ to 46.88 GPa at Al₁₁La₃ and then slowly decreasing to 30.78 GPa. The Young’s modulus E and shear modulus G of aluminum are 76.11 GPa and 28.30 GPa, respectively. It is obvious that the addition of RE elements makes the Young’s modulus E and shear modulus G of the aluminum alloy higher than that of aluminum, except for Sc. Among them, Al₁₁La₃ has the highest Young’s modulus E value and shear modulus G value, which are greatly improved compared with aluminum, which are increased by about 52% and 66%, respectively. After adding rare earth elements (except Sc), the stiffness of the aluminum alloy material increases, the toughness is good, the material is not easily deformed, and the tensile strength is significantly improved. In contrast, the shear modulus G of Al₁₁Sc₃ is about 18% lower than that of aluminum, and the Young’s modulus E is about 16% lower than that of aluminum, indicating that the element Sc does not enhance its elastic properties, and that the formed aluminum-based alloy is volatile. Moreover, stiffness and strength decrease, and elastic deformation increases.

Poisson’s ratio ν can be used to predict the plasticity of the material, thereby quantifying the stability of the crystal against shearing [20,46]. It also represents the ratio of the lateral strain to the longitudinal strain of the solid medium; therefore, the range of Poisson’s ratio is greater than 0, the Poisson’s ratio is in the range of 0.1~0.3, and the material is hard; when Poisson’s ratio is 0.4~0.5, the material is generally fluid or soft. Figure 2c shows that Al₁₁La₃ has the lowest Poisson’s ratio of 0.24, which is also in the range of 0.1 to 0.3, which belongs to an alloy with high hardness. Generally speaking, aluminum has low hardness, high plasticity, and is easy to process. It can be seen from Figure 2c that the Poisson’s ratios of Al, Al₁₁Sc₃, and Al₁₁Er₃~Al₁₁Lu₃ are all between 0.3 and 0.4, with certain plasticity and low hardness. At the same time, Poisson’s ratio v can reflect the heterogeneity of materials. Taking the Poisson’s ratio v = 0.26 as the boundary, except for Al₁₁Ce₃ and Al₁₁La₃, all aluminum-based alloys exhibit uneven deformation, and the Poisson’s ratio v of Al₁₁Sc₃ (0.37) is the largest, and the heterogeneity is obvious.

B/G can also be used to characterize the plasticity of the material: with a B/G of 1.75 as the boundary [47], below 1.75, the material is ductile, and, above 1.75, the material is brittle [20,45]. As shown in Figure 2c, the Poisson’s ratio v and B/G of the aluminum alloy both decrease first and then increase with an increase in the rare earth atomic number, taking the La element as the node. Among them, the B/G of Al₁₁La₃, Al₁₁Ce₃, and Al₁₁Pr₃ are all lower than 1.75, with values of 1.56, 1.67, and 1.71, respectively, showing certain ductility. The value of Al₁₁La₃ is the smallest, which indicates that the ductility is good and that the rolling degree is improved, which is beneficial to the production of precision materials. The B/G value of Al₁₁Sc₃ is up to 3.567, which is not conducive to rolling.

3.3. The Thermodynamic Properties

In order to understand the thermodynamic properties of the second-phase particles of the material, this calculation uses the Debye theory (DT) [35] to obtain a model of crystal vibration [48,49,50], namely, the Debye–Grüneisen model, and the key is to calculate the Debye function ${\Theta }_D$(V) through the Debye cutoff frequency ${\omega }_D$ [51,52,53]. At high temperature [54,55,56], the Al₁₁RE₃ phase will be decomposed into Al and Al₂RE; in this work, we adopt P = 0 at a temperature of 0–1000 K [57] for the thermodynamic calculation of the aluminum alloy, according to the following equation [51,52,53]:

$$\frac{h}{2\pi }{\omega }_D={\kappa }_B{\Theta }_D$$

(15)

where the h and ${\kappa }_B$ are Plank’s and Boltzmann’s constants, respectively. The ${\omega }_D$ can be obtained using the Debye sound velocity $v_D$:

$${\omega }_D={\left(\frac{6{\pi }^2}{V}\right)}^{1/3}{v}_D$$

(16)

where the V is the volume of the solid. There are three different types of sound velocity in solids, all of which are anisotropic. For isotropic media, Anderson expresses the low temperature average for the sound velocity for an isotropic crystal as:

$$\frac{1}{v_D^3}=\frac{1}{3}\left(\frac{1}{v_L^3}+\frac{2}{v_S^3}\right)$$

(17)

where $v_L$ and $v_S$ are longitudinal and transverse sound velocities and can be expressed as $v_L=\sqrt{\left(L/\rho \right)}$ and $v_S=\sqrt{\left(S/\rho \right)}$, respectively. With Equations (14)–(17), a simpler method can be obtained to calculate $v_D$ and ${\Theta }_D$; then, they are used to calculate the longitudinal and transverse moduli as follows:

$$L=\frac{3\left(1-v\right)}{1+v}B, S=\frac{3\left(1-2v\right)}{2\left(1+v\right)}B$$

(18)

Then, the $v_D$ can be obtained:

$$v_D=f\left(v\right)\sqrt{\frac{B}{\rho }}$$

(19)

$$f\left(v\right)={\left\{\frac{1}{3}\left[{\left[\frac{1+v}{3\left(1-v\right)}\right]}^{3/2}+2{\left[\frac{2\left(1+v\right)}{3\left(1-2v\right)}\right]}^{3/2}\right]\right\}}^{-1/3}$$

(20)

It has been proved that the method can obtain reasonable Debye sound velocity. Therefore, the ${\Theta }_D$ can be calculated from the bulk modulus and density using the following equation:

$${\Theta }_D=f\left(v\right)\frac{h}{2\pi {\kappa }_B}{\left(48{\pi }^5\right)}^{1/6}\sqrt{\frac{r_{0}B}{M}}$$

(21)

where M is the atomic weight and r0 is the equilibrium Wigner–Seitz radius.

After inserting the ${\Theta }_D$ into quasi harmonic formula, the Helmholtz free energy F, entropy S, heat capacity CV, and thermal expansion α can be, respectively, described by [34,53]:

$$F\left(V,T\right)=E\left(V\right)-{\kappa }_{B}T\left\{D\left(\frac{{\Theta }_D}{T}\right)-3\ln \left[1-e^{\left(\frac{-{\Theta }_D}{\mathrm{T}}\right)}\right]+\frac{9}{8}{\mathsf{\kappa }}_{\mathrm{B}}{\Theta }_D\right\}$$

(22)

$$C_V=3{\kappa }_B\left[4D\left({\Theta }_D/T\right)-\frac{3{\Theta }_D/T}{exp\left({\Theta }_D/T\right)-1}\right]$$

(23)

$$S=3{\kappa }_B\left\{\frac{4}{3}D\left({\Theta }_D/T\right)-\ln \left[1-exp\left(-{\Theta }_D/T\right)\right]\right\}$$

(24)

$${\gamma }_D=-\frac{ln\partial {\Theta }_D}{ln\partial lnV}, \alpha =\frac{\gamma C_V}{B_{0}V}$$

(25)

At a temperature of 0–1000 K, the Gibbs free energy F, entropy S, thermal melt C_V, and thermal expansion coefficient $\alpha$ are shown in Figure 3a–d.

As shown in Figure 3a–c, it is found that the thermal expansion coefficient α, constant volume specific heat C_V, and entropy S of the aluminum and rare earth aluminum alloys all increase with an increase in temperature of ~1000 K; however, the increase range is different.

The thermal expansion coefficient α can be used to characterize the ability of an object to expand and contract due to temperature changes. In Figure 3a, the initial and final values of thermal expansion coefficients and growth rates of 17 Al₁₁RE₃ alloys are basically the same, i.e., in the order of ~2.89 × 10⁻⁵/K. In detail, at 0–200 K, there is a relatively large increase, and, after 200 K, the growth rate is slow. The change of thermal expansion coefficient α of an Al of ~5.00 × 10⁻⁵/K with T is significantly larger than that of Al₁₁RE₃, indicating that the thermal stability of the aluminum element is poor, and that the addition of rare earth elements greatly improves the thermal stability of the aluminum element. Aluminum alloy can maintain a good stable state even at high temperature, suitable for high temperature resistant aluminum alloy materials.

As shown in Figure 3b, the constant volume specific heat C_V of Al and Al₁₁RE₃ both showed a significant increase, then a slow increase, and, finally, a gradual increase. The constant volume specific heat C_V of RE aluminum alloys grows the most at 0–200 K, increasing exponentially, and then the growth rate slows down significantly at 200–600 K, and basically stabilizes after 600 K. The specific heat C_V value Al₁₁RE₃ of ~24.86 J/mol/K and the growth rate are both higher than those of the Al of ~24.81 J/mol/K. The growth rates of rare earth aluminum alloys are basically the same, and the constant volume specific heat C_V of Al₁₁RE₃ tends to be consistent. Since the constant volume specific heat C_V of an RE aluminum alloy is greater than that of aluminum, it means that, at a certain temperature, the heat absorption or heat dissipation ability of an aluminum alloy after adding RE elements is stronger than that of aluminum.

The entropy S value indicates that the system changes from an ordered state to a disordered state. The entropy S value increases with temperature because molecular motion becomes more violent with an increasing temperature and will deviate more from its equilibrium position. Looking at Figure 3c, the entropy S curves of aluminum of ~61.27 J/mol/K and aluminum alloys of ~66.86 J/mol/K have been increasing with an increase in temperature; moreover, although the growth rate has decreased, the growth rate and growth value are higher than those of aluminum. The entropy growth and growth trend among the 17 kinds of aluminum alloys are very close, indicating that the addition of RE elements does not mean that the aluminum alloys are greatly affected by temperature and have good consistency.

As shown in Figure 3d, it is found that the Gibbs free energies F of both aluminum and Al₁₁RE₃ alloys are less than 0, which have an Al value of ~−3.712 eV/atom and Al₁₁RE₃ alloys of ~4.137 eV/atom, indicating that they exist stably. At 0 and 1000 K, the Gibbs free energies of lanthanides are also very close, while the initial and final values of Sc and Y are smaller than those of lanthanides. It is also clearly observed that the variation curves of the Gibbs free energy F of the aluminum alloy with the temperature T all decrease, and that the decreasing range is consistent. At the same time, the Gibbs free energy of 17 RE aluminum alloys is lower than that of aluminum, which means that the addition of rare earth elements improves the stability of aluminum.

The variation of the thermal expansion coefficient a of the aluminum element with the temperature T is significantly larger than that of Al₁₁RE₃, indicating that the thermal stability of the aluminum is poor, and that the addition of rare earth elements greatly improves the thermal stability of the aluminum alloy, making the aluminum alloy exist even at high temperatures. Nevertheless, it can still maintain good stability in the state and is suitable for high temperature-resistant aluminum alloy materials.

4. Conclusions

In summary, the stability, mechanical properties, and thermodynamic properties of eleven Al₁₁RE₃ intermetallics at 0–1000 K were investigated by combining the first-principles calculations and the Debye model. The results of the study are as follows:

(1) The enthalpy of formation $\Delta H_f$ of Al₁₁RE₃ intermetallics are all negative values, all belong to exothermic reactions, and are all stable compounds. The thermodynamic stability of Al₁₁La₃ is the strongest, while the thermodynamic stability of Al₁₁Sc₃ is the weakest;

(2) Comparing the mechanical constants of the eleven Al₁₁RE₃ intermetallics, it is found that the stiffness, toughness, and tensile strength of the Al₁₁RE₃ intermetallics (RE = La, Ce, Pr, Nd, Sm) are significantly greater than those of aluminum, which indicates that these Al₁₁RE₃ intermetallics act as strengthen phases that can effectively improve the mechanical property of aluminum;

(3) The Debye–Grüneisen model was used to analyze the thermodynamic properties (Gibbs free energy, constant volume specific heat, entropy, and thermal expansion coefficient) of the eleven Al₁₁RE₃ intermetallics at a range of 0–1000 K, and the results consistently showed that the Al₁₁RE₃ intermetallics act as strengthen phases that can improve the thermal stability of the aluminum. In the process of temperature change, Al₁₁Pm₃ and Al₁₁Sm₃ are most affected by temperature and easily become unstable; their heat absorption and heat dissipation capabilities are also stronger than other RE aluminum alloys;

(4) With an increase in the atomic number, different change trends were observed for the formation enthalpy $\Delta H_f$, bulk modulus B, Young’s modulus E, and shear modulus G, which decreases first.